My question is the following:
Is the borel sigma algebra (field) $\mathcal{B}(\mathbb{R}^n)$ generated by the set $$\mathcal{A} :=\{E_1 \times E_2 \times \cdots \times E_n : E_1 \in \mathcal{B}(\mathbb{R}), ...,E_n\in \mathcal{B}(\mathbb{R}) \}?$$ In other words, if $\sigma(\mathcal{A})$ denotes the sigma field generated by $\mathcal{A}$ (i.e. the smallest sigma field containing $\mathcal{A}$). Is then $\sigma(\mathcal{A})=\mathcal{B}(\mathbb{R}^n)$? And if so, how can I prove (or see) that?
Thanks in advance!